# Finite Elements for Higher Order Steel–Concrete Composite Beams

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## Abstract

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## Featured Application

**The paper presents finite elements for a higher order steel–concrete composite beam model that can be implemented within commercial software for structural analysis and used for the design of composite bridges.**

## Abstract

## 1. Introduction

^{®}[32].

## 2. Recall of the Analytical Model

#### 2.1. Kinematics

#### 2.2. Active Stress Field

**I**

_{α}and

**I**

_{s}appearing in Equations (14)–(15) are inertia terms of the cross section of the beam components, as detailed in Appendix A. It is worth mentioning that active stresses of Equations (9)–(11), i.e., relevant to strains descending from the admissible displacement field according to constitutive relationships, do not satisfy the local equilibrium, which also requires additional non-vanishing stress components, called reactive stresses. The latter do not appear in the virtual work theorem expression and can be estimated by means of the local equilibrium conditions. These components are significant in the case of shear stresses, while normal stresses can be obtained with negligible errors from Equations (9)–(11). Expressions of the total shear stresses ${\tau}_{cxz}$ and ${\tau}_{s\xi z}$ are available in Appendix B.

#### 2.3. Balance Conditions

#### 2.4. Analytical Solution

#### 2.4.1. Beam with Imposed End Displacements

#### 2.4.2. Beam with Distributed Loads

## 3. Proposed Higher Order Finite Elements

#### 3.1. Definition of the Beam Finite Elements

#### 3.1.1. Linear Interpolating Functions

#### 3.1.2. Polynomial Interpolating Functions

#### 3.1.3. Interdependent Interpolating Functions

#### 3.2. Convergence Analysis of the Proposed Finite Elements

## 4. Numerical Comparison with More Complex Models

^{®}[32]. For this purpose, all the analysis cases previously presented, including loading conditions and static schemes, are investigated.

## 5. Conclusions

- The finite element based on linear shape functions (GFE) suffers from locking problems and requires a highly refined discretization to reach an accurate solution of the problem;
- The finite element implementing cubic and quadratic polynomial shape functions (CIFE) avoids locking problems and is characterised by a higher converge rate than that based on linear shape functions (GFE);
- The finite element with exponential shape functions (IIFE) is the most performant and furnishes an almost exact solution, independent of the beam discretization, provided that enough finite elements are adopted to avoid issues in the numerical evaluation of the exponential matrix;
- The CIFE is highly competitive with respect to the IIFE, especially in predicting beam displacements and rotations; in some cases, if very accurate solutions are not required, the former may provide results with a lower number of finite elements than that necessary to avoid instabilities in the computation of the exponential shape functions of the IIFE;
- In the case of distributed or concentrated loads, the convergence rate relevant to warping intensities of the steel components is much lower than that relevant to the other response parameters; differences in the convergence rate attenuate in the cases of prestressing or concrete shrinkage.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Stress Resultants and Inertial Components

## Appendix B

#### Stress State

## Appendix C

#### Notations

0 | origin of Cartesian coordinate system; |

$\mathit{A}$ | matrix containing stiffnesses of the beam cross section; |

$A$ | area; |

$\mathit{a}$ | geometric vector; |

$\mathit{B}$ | matrix containing stiffnesses of the beam cross section; |

$\mathit{b}$ | vector of the integration constants; |

$B$ | concrete slab width; |

$\mathit{C}$ | matrix containing stiffnesses of the beam cross section; |

$\mathit{c}$ | vector of loads and stress-independent strain along the beam; |

$\mathit{d}$ | vector of all unknown displacements; |

$d$ | differential operator; |

$\mathit{E}$ | exponential matrix; |

$E$ | Young’s modulus; |

$\mathit{f}$ | vector of nodal forces; |

${f}_{c},{f}_{sh},{f}_{sv}$ | warping intensity functions; |

$G$ | shear modulus; |

${h}_{c}$ | slab thickness; |

$\mathit{I}$ | inertia matrix or identity matrix; |

$\mathit{J}$ | inertia matrix; |

$\mathit{K}$ | stiffness matrix of the beam element; |

$k,i$ | indexes; |

$L$ | length of the beam; |

$l$ | length of the beam plane walls; |

${l}_{e}$ | length of the finite element; |

$\mathit{L}$ | inertia matrix; |

$M$ | bending moment at the beam end cross section; |

$m$ | bending moment along the beam axis; |

$N$ | longitudinal force at the beam end cross section; |

${\mathit{N}}_{e}$ | matrix of interpolating functions; |

$\overline{\mathit{n}}$ | vector of resultants of forces due to restrained stress-independent strain; |

$n$ | number of the plane steel walls; |

${n}_{e}$ | number of finite elements |

$\mathit{p}$ | resultants of external forces along the beam axis; |

$\mathit{P}$ | resultants of external forces at the beam end cross section; |

${Q}_{v}$ | resultant of vertical loads at the beam end cross section; |

${q}_{c},{q}_{s}$ | longitudinal forces along the beam axis; |

${q}_{v}$ | resultant of vertical loads along the beam axis; |

$\mathit{R}$ | inverse of matrix of exponential matrices evaluated at beam ends; |

$\mathit{s}$ | vector grouping unknown displacements and their first derivative; |

${t}_{i}$ | thickness of the i-th plane steel wall; |

$\cup $ | linear matrix operator; |

$\mathit{U}$ | displacement of the two end cross sections of the beam; |

$\mathit{u}$ | displacement of the end cross section of the beam; |

$u$ | transverse displacement, along coordinate direction $X$; |

${\mathit{v}}_{e}$ | vector of the unknown nodal displacements; |

$\mathit{v}$ | assembled vector of the nodal displacements of all the elements; |

$v$ | vertical displacement of the cross section, along coordinate direction $Y$; |

$W$ | bi-moment at the beam end cross section; |

$w$ | longitudinal displacement, along coordinate direction $Z$; |

$\mathit{w}$ | vector grouping the generalised displacements; |

$X,Y,Z$ | coordinate axes; |

$x,y,z$ | coordinates; |

$\overline{x},\overline{y}$ | coordinates of the slab–girder interface connection; |

$\alpha $ | direction cosine of the local abscissa; |

$\u03f5,\text{}k,\text{}\mu $ | overall stress-independent strain; |

$\overline{\mathit{\epsilon}}$ | vector of stress-independent strains; |

$\overline{\epsilon}$ | stress-independent strain; |

$\tilde{\epsilon}$ | generic nonlinear stress-independent longitudinal strain field; |

$\Phi $ | rotation; |

$\Gamma $ | beam–slab interface slip; |

$\eta $ | local abscissa of the finite element; |

$\lambda $ | normalised abscissa of the finite element |

$\mu $ | interpolating function; |

$\nu $ | Poisson’s ratio; interpolating function |

$\rho $ | stiffness per-unit-length of the shear connection; |

${\sigma}_{z}$ | normal stress; |

$\tau $ | shear stress; |

$\upsilon $ | interpolating function; |

$\omega $ | bi-moment along the beam axis; |

$\xi $ | local abscissa of the beam plane walls; |

${\psi}_{c}$ | slab warping function; |

${\psi}_{sh}$ | steel warping function due to longitudinal shear flow; |

${\psi}_{sv}$ | steel warping function due to shear force. |

Subscripts | |

$c$ | concrete part of the composite beam; |

$e$ | finite element; |

$r$ | steel reinforcement part of the composite beam; |

$s$ | steel part of the composite beam; |

$0$ | referred to the origin of coordinate system; |

$,$ | partial derivatives. |

Symbols and Superscripts | |

$T$ | concrete part of the composite beam; |

${}^{\prime}$ | derivative with respect to z variable; |

$\mathcal{D}$ | formal linear differential operator; |

$^$ | variation; |

$\cdot $ | scalar product. |

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**Figure 1.**Examples of steel–concrete bridge decks: (

**a**) continuous twin-girder multi-span viaduct-Serra Cazzola viaduct, AG, Italy; (

**b**) continuous box-girder multi-span bridge on the Cesano River, at Mondolfo, PU, Italy.

**Figure 3.**(

**a**) Nodal generalized displacements of the general finite element (GFE), (

**b**) interpolation along its abscissa.

**Figure 4.**(

**a**) Nodal generalized displacements of the consistent interpolation finite element (CIFE); (

**b**) interpolation of longitudinal displacement, warping intensities, and rotations; and (

**c**) interpolation of vertical displacement.

**Figure 6.**(

**a**) Load conditions and (

**b**) static schemes considered in the applications. UDL, uniformly distributed load; CL, concentrated load; SS, concrete slab shrinkage; PW, prestressing of wires in the concrete slab.

**Figure 7.**S2-UDL: Normalized beam response quantities for the (

**a**) GFE and (

**b**) CIFE; (

**c**) errors with respect to the reference solution.

**Figure 8.**S3-CL: Normalized beam response quantities for the (

**a**) GFE and (

**b**) CIFE; (

**c**) errors with respect to the reference solution.

**Figure 9.**S2-SS: Normalized beam response quantities for the (

**a**) GFE and (

**b**) CIFE; (

**c**) errors with respect to the reference solution.

**Figure 10.**S3-PR: Normalized beam response quantities for the (

**a**) GFE and (

**b**) CIFE; (

**c**) errors with respect to the reference solution.

**Figure 11.**Trend of the coefficients of variation (COVs) for all the case studies and beam response quantities.

**Figure 13.**Comparison between the proposed beam model and the refined 3D FE model for some case studies in terms of vertical displacement ${v}_{0}$; interface slip ${\Gamma}_{z}$; and the longitudinal displacement of the steel girder bottom and upper flanges, ${w}_{s,inf}$ and ${w}_{s,sup}$.

**Figure 14.**(

**a**) Longitudinal normal stress on the slab mid plane, (

**b**) on the steel girder web, and (

**c**) on the steel girder bottom flange for case study S2-UDL.

**Figure 15.**(

**a**) Longitudinal normal stress on the slab mid plane, (

**b**) on the steel girder web, and (

**c**) on the steel girder bottom flange for case study S3-CL.

**Figure 16.**(

**a**) Longitudinal normal stress on the slab mid plane, (

**b**) on the steel girder web, and (

**c**) on the steel girder bottom flange for case study S2-SS.

**Figure 17.**(

**a**) Longitudinal normal stress on the slab mid plane, (

**b**) on the steel girder web, and (

**c**) on the steel girder bottom flange for case study S3-PR.

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**MDPI and ACS Style**

Gara, F.; Carbonari, S.; Leoni, G.; Dezi, L.
Finite Elements for Higher Order Steel–Concrete Composite Beams. *Appl. Sci.* **2021**, *11*, 568.
https://doi.org/10.3390/app11020568

**AMA Style**

Gara F, Carbonari S, Leoni G, Dezi L.
Finite Elements for Higher Order Steel–Concrete Composite Beams. *Applied Sciences*. 2021; 11(2):568.
https://doi.org/10.3390/app11020568

**Chicago/Turabian Style**

Gara, Fabrizio, Sandro Carbonari, Graziano Leoni, and Luigino Dezi.
2021. "Finite Elements for Higher Order Steel–Concrete Composite Beams" *Applied Sciences* 11, no. 2: 568.
https://doi.org/10.3390/app11020568